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Full text of "Treatise On Analysis Vol-Ii"

402 XV NORMED ALGEBRAS AND SPECTRAL THEORY In particular: (15.11.12) If H is any positive, self-adjoint operator then there exists a unique-positive self-adjoint operator H1 such that H'2 = H. Apply (15.11.11) with M = N = R+ and/(C) = C2. The unique positive self-adjoint operator H' defined in (15.11.12) is denoted by H112. Example (15.11.13) Let E be the Hilbert space Lc(R; A), where A is Lebesgue measure. Since the function /(£) = e~^ *s A-integrable, the convolution g*-*f*g defines, on passing to the equivalence classes, a continuous operator H on E (14.10.6) with norm N^/) =2. As in (11.6.1), it is immediately seen that H is self-adjoint. It can be shown directly (Problem 5) that the interval [0, 2] in R is equal to Sp(H); this also follows from the general theorems of harmonic analysis (Chapter XXII). Note that, for each a e R, the function ga(t) = eiat is such that the con- volution/*^ is defined and equal to 2gJ(l + a2). However, it is not the case that the ga are " eigenfunctions " of H, because they do not belong to &c(R, A). In Chapter XXIII we shall obtain a generalization and an interpre- tation of this phenomenon. (15.11.14) The case of normal operators whose spectrum contains no non- isolated point 7^0. In this case (which is that of compact normal operators (11.4.1)) let (An) be the (finite or infinite) sequence of points of Sp(N), other than 0. These are the eigenvalues of N (15.11.6). The eigenspace E(AW; N) corresponding to AM is just the space E({AW}) defined in (15.11.9), and these closed subspaces are therefore pairwise orthogonal. Moreover, we have E({0}) = Ker(N), the spectrum of the restriction of N to this subspace being reduced to 0 (15.11.9). Finally, E is the Hilbert sum of E({0}) and the E({AW}). For it is enough to apply (15.11.8(iv)) to the sequence of functions (/„), where /n(Q =1 for £ = Ak with k£n, /„(£) = 0 for C = Ak and k > n, and /n(0) = 1; this shows in particular that every x e E is the limit of the sum of its projections on E({0}) and the E({Ak}) with k <£ n. Hence the result (6.4). In particular, if E is ./mfte-dimensional, a normal operator on E may be defined to be an operator whose matrix is diagonal, with respect to a suitably chosen orthonormal basis of E.must hav0^^ ^ which is the inverse of/>then bY